Given: Ry uses his AR-15 to shoot 1200 grams of Boomerite contained in a coffee creamer container. On top of the coffee creamer container is a 60 pound steel contraption for crushing charcoal briquettes and launching the dust into the air. Joe takes a video using his Windows Phone 7 phone and puts it up on YouTube*. In the video you can see the explosion occurred at 11.18 seconds into the video. The charcoal dispenser hits the ground at 14.48 seconds into the video. Afterward Ry measures the horizontal distance the charcoal dispenser traveled. It is 13 yards. Assume the acceleration of gravity on this planet at this location is 32.174 ft/sec2.
Problem: Ignoring air resistance and assuming the initial acceleration was for all practical purposes instantaneous answer the following questions:
- How high into the air did the charcoal dispenser go?
- At the instant after the detonation what were the horizontal and vertical velocity vectors of the charcoal dispenser?
- At the instant after the detonation what was the total velocity vector of the charcoal dispenser?
- What was the USPSA power factor of the charcoal dispenser at launch?
- If used at an USPSA match does the charcoal dispenser “make Major” for both pistol and rifle competition?
Be sure to use consistent units during the calculations and give the results in English units.
- The total time in the air is 3.3 seconds. One half of the time is spent going up and the other half is spent going down. The equation of motion for an object dropped in a gravitational field is:
d = 1/2 a t2
Where d is the distance traveled in feet, a is the acceleration of the gravitational field, and t is the time in seconds.
The maximum height can be expressed as:
d = (32.174/2 ft/sec2) (3.3 sec/2)2
d = (16.087 ft/sec2)(1.65 sec)2
d = (16.087 ft/sec2)(2.7225 sec2)
d = 43.8 ft
- The equation of motion for an object traveling at a constant speed is:
d = v t
Where d is the distance traveled, v is the velocity, and t is the time.
This can be used to give us the initial horizontal velocity component.
Since the total time in the air was 3.3 seconds and the horizontal distance traveled was 13 yards the velocity can be solved for in the following equation:
13 yards = (v)(3.3 sec)
v = (13 yards)/(3.3 sec)
v = 3.94 yards/sec
or expressed in the more common feet per second:
v = (3 ft/yard)(3.94 yards/sec)
v = 11.8 ft/sec
The vertical component at launch is the same as the final vertical velocity at the moment of impact. The equation of velocity with respect to time is:
v = a t
Where v is the final velocity, a is acceleration, and t is the time.
Hence the initial vertical velocity is:
v = (32.174 ft/sec2)(3.3/2 sec)
v = (32.174 ft/sec2)(1.65 sec)
v = 53.1 ft/sec
- The total velocity is the square root of the sum of the squares of the horizontal and vertical velocities. Hence the total velocity at the instant after detonation was:
v = SQRT((11.8 ft/sec)2 + (53.1 ft/sec)2)
v = 54.4 ft/sec
- IPSC Power Factor is expressed by the following equations
PF = (m v)/1000
Where m is the mass of the bullet in grains and v is the velocity of the bullet in ft/sec.
There are 7000 grains in one pound. Hence the mass of the “bullet” is (7000)(60) or 420,000 grains.
Hence the IPSC Power Factor is:
PF = (420,000)(54.4)/1000
PF = 22,848
- The minimum USPSA power factor required to make major with a pistol is 165. For rifle it is 320. Since 22,848 is greater than both 165 and 320 the answer is “Yes”.
* The YouTube video: